We study a class of second-order elliptic equations of divergence form, withdiscontinuous coefficients and data, which models the conductivity problem incomposite materials. We establish optimal gradient estimates by showing theexplicit dependence of the elliptic coefficients and the distance betweeninterfacial boundaries of inclusions. The novelty of these estimates is thatthey unify the known results in the literature and answer open problem (b)proposed by Li-Vogelius (2000) for the isotropic conductivity problem. We alsoobtain more interesting higher-order derivative estimates, which answers openproblem (c) of Li-Vogelius (2000). It is worth pointing out that the equationsunder consideration in this paper are nonhomogeneous.
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